## 3. Higher order statistics via the wavelet coefficients## 3.1. DWT decomposition and 2nd order statisticsIt is generally believed that the cosmic mass (or number) density distribution can be mathematically treated as a homogeneous random field. It is often more convenient to express in terms of its Fourier transform, , which is the Fourier coefficient of . One reason for expressing the density distribution in terms of its Fourier transform is that for Gaussian random fields all the statistical features of can be completely described by the amplitude of the Fourier coefficients. In this case, the phase of is not important and the power spectrum of the perturbations, , or equivalently, the two-point correlation function, are all that is necessary to describe the statistical behavior of the density distribution. However, if the field is non-Gaussian, then in order to have a full statistical description of the field the phases of the Fourier coefficients are essential. As is well known, it is difficult, even practically impossible, to
find information about the phases of the Fourier coefficient as soon
as there is some computational noise. The lack of information about
the phases makes the description incomplete: we
might know the scales Let's consider a 1-D mass density contrast , which covers a spatial range . The expansion of the field in terms of the DWT basis is given by where , , are the basis of the DWT. The DWT basis are orthogonal and complete. The wavelet function coefficient (WFC), , is computed by The wavelet transform basis functions are
generated from the basic wavelet by a dilation,
, and a translation The basic wavelet is designed to be
continuous, admissible and localized. Unlike the Fourier basis
, which are non-local in physical space, the
wavelet basis are localized in both physical
space and Fourier (scale) space. In physical space,
is centered at position ,
and in Fourier space, it is centered at wavenumber
. Therefore, the WFCs, ,
have two subscripts A clearer picture of how the transforms work can be seen in the
phase space . A complete, orthogonal basis set
resolves the whole phase space into "elements" of size
and . Each mode
corresponds to elements in the phase space. For the Fourier transform,
this corresponds to elements of size and
. For the wavelet transform, both
and are finite, and the
corresponding area of the element is as small as
. That is, the DWT is able to resolve an
arbitrary function simultaneously in terms of The WFC and its intensity describe, respectively, the fluctuation of the density and its power on scale at position . As with the Fourier basis, Parseval's theorem holds for the DWT basis. It is (Fang & Pando 1997, Pando & Fang, 1998) It is possible to define the power spectrum of the density
perturbation on scale where It has been shown that the DWT power spectrum Eq. (6) can be converted to the Fourier power spectrum, i.e. in terms of second order statistical description the DWT and Fourier transform are equivalent. ## 3.2. One-point distribution of WFCs and non-GaussianityThe cosmic density field is usually assumed to be ergodic: the
average over an ensemble is equal to the spatial average taken over
one realization. This is the so-called "fair sample hypothesis"
(Peebles 1980). A homogeneous Gaussian field with continuous spectrum
is certainly ergodic (Adler 1981). In some non-Gaussian cases, such as
homogeneous and isotropic turbulence (Vanmarke, 1983), ergodicity also
approximately holds. Roughly, the ergodic hypothesis is reasonable if
spatial correlations are decreasing sufficiently rapidly with
increasing separation. In this case, the volumes separated by large
distances are approximately statistically independent, and can be
treated as independent realizations. Note that the
are orthogonal with respect to the position
index Consequently, the distribution of the is
actually the one-point distribution of the WFCs at a given scale
where From Eqs. (6), (7) and (8), one sees that the second order cumulant
moment is the DWT power spectrum on the scale For Gaussian fields all the cumulant moments higher than order 2
are zero. Thus one can measure the non-Gaussianity of the density
field by with
. Analogous to being
called the DWT power spectrum, we will call the
DWT spectrum of and are basic statistical measures employed in this paper. For comparison, the definitions of the "standard" skewness and where the variance is given by Obviously, no scale information is given by ## 3.3. Central limit theorem and the DWT analysisIt is well known that not all one-point distributions can detect non-Gaussianities. This is due to the constraints imposed by the central limit theorem. For instance, if the universe consists of a large number of dense clumps with a non-Gaussian probability distribution function (PDF), the one-point distributions of the real and imaginary components of each individual Fourier mode are still Gaussian due to the central limit theorem (Ivanonv & Leonenko 1989). Further, even when the non-Gaussian clumps are correlated the central limit theorem still holds if the two-point correlation function of the clumps approaches zero sufficiently fast (Fan & Bardeen 1995). For these reasons, the one-point distribution function of Fourier modes is not sensitive enough to detect deviations from Gaussian behavior. Even for samples with strong non-linear evolution, the one point distribution function of Fourier modes is found to be consistent with a Gaussian distribution (Suginohara & Suto 1991). It should be pointed out that the inefficiency of the Fourier mode one-point distribution in detecting non-Gaussianity is not because Fourier transform loses information about the distribution . The Fourier coefficients contain all the information on non-Gaussianity, but the information is mainly contained in the phases of the Fourier coefficients. As mentioned in Sect. 3.1, it is very difficult to detect the distribution of the phases of Fourier coefficients. On the other hand, the wavelet coefficients are not subject to the
central limit theorem. In this respect, the DWT analysis is similar to
the count in cell (CIC) method (Hamilton 1985; Alimi, Blanchard &
Schaeffer 1990; Gaztañaga & Yokoyama 1993; Bouchet et al.
1993; Kofman et al. 1994; Gaztañaga & Frieman 1994). The
CIC is not subject to the central limit theorem as it based on
spatially localized window functions (Adler 1981). The DWT's basis are
also localized. If the scale of a clump in the universe is This point can also be shown from the orthonormal basis being used
for the expansion of the distribution . A basic
condition of the central limit theorem is that the modulus of the
basis be less than , where Because the magnitude of the basic wavelet
is of order 1. The condition will no longer
hold for a constant © European Southern Observatory (ESO) 1998 Online publication: November 9, 1998 |